Why Cepheids have Period Luminosity relation?

Question

According to my astronomy teacher, Cepheids is a type of variable stars that has Luminosity-Period relationship:

$M \propto log(T)$ , where $T$ is the pulsation period of Cepheids.

But I have a question regarding this, there are a lot of variable stars in the universe, why only Cepheids shows a Luminosity-Period relationship while other variables don't show it, does it have something to do with the properties of Cepheids?

On the Wikipedia page about Cepheids, I do see that according to κ model:

$Tρ = \sqrt{C}$ , where

Where $T$ is the pulsation period and ρ is the density of the star., but I do not see how this helps to explain my question.

I also found a pdf on the internet that explains why variables have luminosity period relation, but I have difficulties to explain that only Cepheids and some type of variables have luminosity period relation.

Answer 1

The timescale of any large scale radial variation in a star will, even on a dimensional basis, be proportional to $(G\bar{\rho})^{-1/2}$, where $\bar{\rho}$ is the average density. Thus $$T \propto M^{-1/2}R^{3/2}\ . $$

Cepheid variables have, very roughly the same mass and surface temperature, but a big range of radii. Since luminosity is $\propto R^2$, then that gives $$T \propto L^{3/4}$$ and a period-luminosity relationship.

The pulsations occur at a particular range of temperatures and radii in the Hertzsprung-Russell diagram called the instability strip. Thus is where the conditions are suitable for a subphotospheric layer of ionised helium. This has the property of increasing its opacity when compressed and heated and this is what drives the pulsations.

Period-luminosity relationships are not unique to Cepheids - they are also present in RR Lyrae variables and several other classes of pulsator.

Of course, to get a period-luminosity relationship, you need the pulsators to have a range of radii whilst other physical variables are more narrowly constrained. Or, if on the main sequence, a range of masses. But if the pulsation mechanism demands a fixed temperature then that isn't always possible.

Answer 2

There is some material by Bill Janesh, Case Western Reserve University

Pulsating variable stars

Bill Janesh points out that in order to have a pulsation there must be a process going on that in both directions is overshooting an equilibrium point.

The overshooting is most difficult to understand/model, it would seem.
Light and heat tend to come into an equilibrium state; that's why it is possible to obtain an exact measurement of a black body spectrum. If something repeats well then you can measure it well. Thermal equilibrium between heat and light repeats well.

However, in the case of variables with a regular pulsation cycle something is thwarting the normal tendency to equilibrium state.

With Cepheid variability:
We have that there is a phase where internal absorption of light exceeds the amount of emission, so that heat accumulates. At some point that accumulation of heat rolls over, and a phase ensues where emission exceeds absorption.

This cyclic unbalance of absorption and emission is attributed to difference of opacity of states of ionization of Helium.

It seems to me that the opacity story is incomplete. If a material becomes more opaque it heats up, and immediately it will start emitting more light, due to its higher temperature.

But what you need is something that causes a lag, some form of hysteresis

Hysteresis effects are difficult to model.

Incidentally, until now I had failed to appreciate that Cepheid variability is a very transient state. A state of Cepheid variability is a state that pretty much every star will enter one or more times during the star's overall lifetime, and it will last for only a very short amount of time (compared to the overall luminous lifetime of the star.

It would seem that in order for Cepheid variablity to occur a lot of transient properties have to line up in a very specific way.

It would seem: a highly specific set of conditions must line up for a star to enter a phase of Cepheid variabiity. It seems that the combination of conditions is so specific that stars in Cepheid phase are lookalikes. That would go towards explaining why there is a strong correlation between period of the Cepheid cycle, and luminosity of the star.